Numbers 5:11-31 gives us an adultery test used by the Hebrew people in their early days.

11 Then the Lord spoke to Moses, saying, 12 “Speak to the sons of Israel and say to them, ‘If any man’s wife goes astray and is unfaithful to him, 13 and a man has intercourse with her and it is hidden from the eyes of her husband and she is undetected, although she has defiled herself, and there is no witness against her and she has not been caught in the act, 14 if a spirit of jealousy comes over him and he is jealous of his wife when she has defiled herself, or if a spirit of jealousy comes over him and he is jealous of his wife when she has not defiled herself, 15 the man shall then bring his wife to the priest, and shall bring as an offering for her one-tenth of an ephah of barley meal; he shall not pour oil on it nor put frankincense on it, for it is a grain offering of jealousy, a grain offering of memorial, a reminder of iniquity.

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29 ‘This is the law of jealousy: when a wife, being under the authority of her husband, goes astray and defiles herself, 30 or when a spirit of jealousy comes over a man and he is jealous of his wife, he shall then make the woman stand before the Lord, and the priest shall apply all this law to her. 31 Moreover, the man will be free from guilt, but that woman shall bear her guilt.'"

I think the "man" in v.31 is referring to the adulterous woman's husband. If so, what guilt is the man 'free' from? If all he has is a righteous jealousy for his wife, then why is there any mention of his guilt? If it's not him, then the verse must be talking about the man who committed adultery with this woman. If so, why is he free from guilt?

2 Answers
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The part of the scripture you left out is pretty important to understanding this passage. Basically, she has to drink some dusty water that will cause her to miscarry if she's guilty, but she'll be fine if she's not guilty.

I think it's simply making clear that the husband is not guilty for miscarriage and the harm that results from the actions of the priest applying the law to his wife. That guilt lies with the woman, because it is her who slept around and lied about it. According to scripture, she brought it on herself. She can't come back and say, "This child would have lived had you not taken me to the priest." In that situation, he could simply respond, "No, the child would have lived had you not had an affair and lied about it."

In other words, he is not held responsible when her actions conflict with God's law, and bring pain and suffering onto her own head, even if he was the one who brought her to the priest for the application of the law.

The spiritual point is this:

The application of the law doesn't bring guilt. The transgression of the law does.

More recent translations use 'husband' instead of 'man' (though not all of them), indicating that it is almost certainly the husband. This is made more likely by the fact that by v31 the 'other party' to adultery hasn't been mentioned in a long while. You can probably ask at Biblical Hermeneutics site for more details.

As for why: he hasn't done anything wrong. His wife committed adultery. It may be that other cultures practices 'guilt by association' in which unwittingly allowing your wife to commit adultery was considered wrong